Quantum mechanics (QM) is a branch of physics that describes the behavior of matter and energy on a microscopic scale, such as atoms and subatomic particles. To study quantum mechanics (In the context of undergraduate level), one must have a solid understanding of mathematics. The following are the mathematical prerequisites for learning quantum mechanics,
1. Linear Algebra: Understanding linear algebra is fundamental to studying QM, as QM heavily relies on the concepts of linear algebra, such as vector spaces, inner products, eigenvalues, and eigenvectors. A basic understanding of finite-dimensional linear algebra is sufficient for introductory QM courses, while infinite-dimensional linear algebra is used in more advanced courses. In the list (May have some overlooked topics),
- Matrix algebra
- Eigenvectors and eigenvalues
- Inner product spaces and orthogonality
- Unitary and Hermitian matrices
- Diagonalization and the spectral theorem
- Linear transformations, rank and nullity, and linear independence
- Tensor product and tensor algebra
- Dirac notation and bra-ket formalism
2. Calculus: A basic understanding of calculus is necessary, specifically derivatives and integrals (I-III). Multivariate calculus is helpful, but not mandatory.
- Limits and continuity
- Differentiation and integration
- Differential equations and solutions
- Taylor and Maclaurin series
- Multivariable calculus and partial derivatives
- Line integrals and Green's theorem
- Stoke's theorem and divergence theorem
3. Probability Theory: Probability theory is used extensively in QM, especially in the interpretation of the wave function. Basic knowledge of probability theory is required to study QM.
- Probability spaces and axioms
- Random variables and distributions
- Conditional probability and Bayes' rule
- Expectation values and variance
- Joint and marginal probabilities
- Normal and Poisson distributions
- Central limit theorem
- Maximum likelihood estimation
4. Complex Numbers (Analysis): Understanding complex numbers is crucial in QM, as the wave function is a complex-valued function. The knowledge of complex analysis is required to study QM.
- Algebraic properties of complex numbers
- Complex functions, including exponential, trigonometric, and hyperbolic functions
- Cauchy-Riemann equations and analytic functions
- Complex integration, including contour integration and Cauchy's theorem
- Power series and Laurent series expansions of complex functions
- Residue theorem and its applications
5. Differential Equations: Basic knowledge of ordinary and partial differential equations is necessary to understand the time-dependent Schrödinger equation, which is a central equation in QM.
- Basic concepts of calculus, including differentiation and integration
- First-order differential equations, including separable and linear equations
- Second-order linear differential equations with constant coefficients
- Laplace transforms and their applications to solving differential equations
- Fourier series and Fourier transforms and their applications to solving differential equations
6. Fourier Analysis: Fourier analysis is used in QM to study the properties of waves and wave functions. Basic knowledge of Fourier analysis is required to study QM.
- Basic concepts of calculus, including differentiation and integration
- Fourier series and their convergence properties
- Fourier transforms and their inverse transforms
- Applications of Fourier analysis, including signal processing and solving differential equations
Some sources suggest that knowledge of group theory and representation theory is also important for QM courses, although they are not required for introductory courses, especially in undergraduate programs. But still, important topics if you want a further understanding of advanced QM.
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Sumber Rujukan :
[1] Quantum mechanics/Course - Wikiversity. https://en.wikiversity.org/wiki/Quantum_mechanics/Course Dicapai 08/03/2023.
[2] What are the mathematical prerequisites to quantum mechanics?. https://matheducators.stackexchange.com/questions/9918/what-are-the-mathematical-prerequisites-to-quantum-mechanics Dicapai 08/03/2023.
[3] What is the math knowledge necessary for starting Quantum Mechanics?. https://physics.stackexchange.com/questions/16814/what-is-the-math-knowledge-necessary-for-starting-quantum-mechanics Dicapai 08/03/2023.
[5] "As stated above QM can be taught and understood at a variety of mathematical levels. At a minimum the required concepts are: 1) Linear Algebra 2) Probability Theory 3) Calculus (basic derivatives and integrals, multivariate would be even better) 4) Newtonian physics and vector mathematics" URL: https://matheducators.stackexchange.com/questions/9918/what-are-the-mathematical-prerequisites-to-quantum-mechanics Dicapai 08/03/2023.
[6] "For good quantum mechanics books, I would recommend: Cohen-Tannoudji, C., Dui, B., Laloe, F., Quantum Mechanics; Feynman, R.P. Hibbs, A., Quantum Mechanics and Path Integrals; Landau, L.D., A Course of Theoretical Physics, Vol. III: Quantum Mechanics; Sakurai, J.J, Modern Quantum Mechanics" URL: https://www.quora.com/What-is-the-mathematical-background-required-to-study-quantum-mechanics?share=1 Dicapai 08/03/2023.
[7] "You will need to understand vector spaces, dual bases, inner products, eigenvalues, etc. on a rigorous level. Group theory and representation theory are definitely not necessary. Although they are important to quantum mechanics, you will not see them until very advanced levels." URL: https://math.stackexchange.com/questions/758502/mathematics-needed-in-the-study-of-quantum-physics Dicapai 08/03/2023.
[8] "Your colleges registrar should tell you which classes are prerequisites for quantum mechanics classes. If you want to know what subjects you have to understand in order to follow quantum mechanics, heres a decent list: 1. Be comfortable manipulating complex numbers. 2. Understand linear algebra." URL: https://www.physicsforums.com/threads/prerequisites-of-quantum-mechanics.486773/ Dicapai 08/03/2023.
[9] "Other prerequisites include the math of linear algebra, Eulers formula from complex analysis and the bra-ket notation. Part 6 Schrödinger Equation Download Article Understand the Schrödinger equation. It is an equation that describes how the quantum state of a physical system changes in time." URL: https://www.wikihow.com/Understand-Quantum-Physics Dicapai 08/03/2023.
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